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arxiv: 2606.31360 · v1 · pith:OXVKBAFTnew · submitted 2026-06-30 · 🧮 math.FA

Bilinear Calder\'{o}n-Zygmund operators on Vilenkin groups

Adil Shafi Wani , Qaiser Jahan , Salman Ashraf This is my paper

Pith reviewed 2026-07-01 03:23 UTC · model grok-4.3

classification 🧮 math.FA
keywords bilinear Calderón-Zygmund operatorsVilenkin groupsL^p boundednessMorrey spacesweak-type estimatesHaar measure
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The pith

Bilinear Calderón-Zygmund operators extend to bounded mappings from L^{p1} × L^{p2} into L^p on Vilenkin groups under the relation 1/p = 1/p1 + 1/p2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that bilinear Calderón-Zygmund operators defined on a Vilenkin group G via suitable kernels are bounded from the product of L^{p1}(G) and L^{p2}(G) into L^p(G) whenever the exponents satisfy the natural relation 1/p = 1/p1 + 1/p2. It first establishes a Grafakos-Torres-type endpoint weak-type result as a preliminary step. The same operators are then shown to map products of Morrey spaces M_{p1,u1}(G) × M_{p2,u2}(G) into M_{p,u}(G) under appropriate conditions on the indices. These statements adapt the classical theory of bilinear singular integrals to the setting of Vilenkin groups with their Haar measure and metric.

Core claim

Bilinear Calderón-Zygmund operators on a Vilenkin group G extend to bounded bilinear mappings from L^{p1}(G)×L^{p2}(G) into L^p(G) under the condition 1/p=1/p1+1/p2, after a Grafakos-Torres-type endpoint weak-type result is proved; the same operators also extend to bounded bilinear mappings from M_{p1,u1}(G)×M_{p2,u2}(G) into M_{p,u}(G) under suitable assumptions on the indices.

What carries the argument

The bilinear kernel satisfying the standard Calderón-Zygmund size, smoothness, and cancellation conditions with respect to the Haar measure and metric on the Vilenkin group G.

If this is right

  • The operators satisfy the full range of L^p boundedness under the exponent relation.
  • An endpoint weak-type inequality of Grafakos-Torres type holds in this setting.
  • The operators satisfy the corresponding boundedness on the indicated Morrey spaces.
  • The classical bilinear estimates carry over directly once the kernel conditions are verified on G.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same kernel conditions could be checked on other locally compact abelian groups to obtain parallel boundedness statements.
  • Endpoint weak-type control might be used to derive further mapping properties at the boundary of the exponent range.
  • The Morrey-space result suggests the operators preserve certain local integrability features on these groups.

Load-bearing premise

The bilinear kernel satisfies the standard Calderón-Zygmund size, smoothness, and cancellation conditions with respect to the Haar measure and metric on the Vilenkin group.

What would settle it

A concrete bilinear kernel on some Vilenkin group that meets the size, smoothness, and cancellation conditions yet fails to be bounded from L^{p1}×L^{p2} into L^p for any choice of exponents satisfying 1/p=1/p1+1/p2 would disprove the main boundedness claim.

read the original abstract

In this article, we study bilinear Calder\'on--Zygmund operators on a Vilenkin group $G$. As a preliminary step, we establish a Grafakos--Torres-type endpoint weak-type result in our setting. Furthermore, we prove that such operators extend to bounded bilinear mappings from $L^{p_1}(G)\times L^{p_2}(G)$ into $L^p(G)$ under the natural condition $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}.$ We then obtain a corresponding boundedness result in Morrey spaces, showing that these operators extend to bounded bilinear mappings from $\mathcal{M}_{p_1,u_1}(G)\times \mathcal{M}_{p_2,u_2}(G)$ into $\mathcal{M}_{p,u}(G)$ under suitable assumptions. These results generalize the classical bilinear estimates to the setting of Vilenkin groups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The paper studies bilinear Calderón-Zygmund operators on a Vilenkin group G. As a preliminary step, it establishes a Grafakos-Torres-type endpoint weak-type result. It then proves that such operators extend to bounded bilinear mappings from L^{p1}(G) × L^{p2}(G) into L^p(G) under the condition 1/p = 1/p1 + 1/p2. Finally, it obtains a corresponding boundedness result in Morrey spaces, showing bounded mappings from M_{p1,u1}(G) × M_{p2,u2}(G) into M_{p,u}(G) under suitable assumptions. These results generalize the classical bilinear estimates to the setting of Vilenkin groups.

Significance. If the results hold, this provides a generalization of bilinear Calderón-Zygmund theory to Vilenkin groups via the Haar measure and group metric. The adaptation is significant for harmonic analysis on non-Archimedean groups. The paper does not ship machine-checked proofs, reproducible code, or parameter-free derivations, but the central claims rest on imposing the standard kernel conditions, after which the boundedness follows by the usual arguments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and the recommendation of minor revision. No specific major comments were provided in the report, so there are no individual points requiring a point-by-point response. We remain available to incorporate any minor editorial changes requested by the editor.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation adapts the classical Grafakos-Torres endpoint weak-type estimate and subsequent L^p and Morrey boundedness arguments to Vilenkin groups by imposing the standard Calderón-Zygmund kernel size/smoothness/cancellation conditions with respect to the Haar measure and group metric. These kernel assumptions are stated as the non-trivial input; the remainder follows by the usual arguments once they hold. No equations reduce a claimed prediction to a fitted parameter by construction, no load-bearing self-citation chains appear, and no ansatz or uniqueness result is smuggled in via prior work by the same authors. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted. The work implicitly relies on the standard Calderón-Zygmund kernel assumptions adapted to the group.

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Reference graph

Works this paper leans on

34 extracted references

  1. [1]

    Springer, Singapore (2021)

    Behera, B., Jahan, Q.: Wavelet analysis on local fields of positive characteristic. Springer, Singapore (2021)

    2021

  2. [2]

    Acta Math.159, 51–80 (1987)

    Christ, M., Journ´ e, J.L.: Polynomial growth estimates for multilinear singular integral opera- tors. Acta Math.159, 51–80 (1987)

    1987

  3. [3]

    Coifman, R.R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc.212, 315–331 (1975)

    1975

  4. [4]

    Springer-Verlag Berlin, Heidelberg, (2012)

    Edwards, R.E., Gaudry, G.I.: Littlewood-Paley and multiplier theory. Springer-Verlag Berlin, Heidelberg, (2012)

    2012

  5. [5]

    Fan, D., Yang, D.: Herz-type Hardy spaces on Vilenkin groups and their applications. Sci. China Math.43(5), 481–494 (2000). 22

    2000

  6. [6]

    Acta Math.129, 137–193 (1972)

    Fefferman, C., Stein, E.M.:H p spaces of several variables. Acta Math.129, 137–193 (1972)

    1972

  7. [7]

    Grafakos, L., Torres, R.H.: Multilinear Calder´ on–Zygmund theory. Adv. Math.165(1), 124–164 (2002)

    2002

  8. [8]

    Shuangping, T.: Estimate for bilinear Calder´ on–Zygmund operators and their commutators on products of variable exponent spaces

    Guanghui, L. Shuangping, T.: Estimate for bilinear Calder´ on–Zygmund operators and their commutators on products of variable exponent spaces. Bull. Korean Math. Soc.59(6), 1471–1493 (2022)

    2022

  9. [9]

    Results Appl

    He, S., Tao, S.: Bilinearθ-type Calder´ on–Zygmund operators and their commutators on gen- eralized weighted Morrey spaces over RD-spaces. Results Appl. Math.26, 100587 (2025)

    2025

  10. [10]

    P-Adic Numbers Ultrametric Anal

    Ho, K.P.: Calder´ on-Zygmund operators on Morrey and Hardy-Morrey spaces in locally compact Vilenkin groups. P-Adic Numbers Ultrametric Anal. Appl.13(3), 204-214 (2021)

    2021

  11. [11]

    Kenig, C.E., Stein, E.M.: Multilinear estimates and fractional integration. Math. Res. Lett. 6(1), 1–15 (1999)

    1999

  12. [12]

    Kitada, T.: Multipliers for weighted Hardy spaces on locally compact Vilenkin groups. Bull. Aust. Math. Soc.48(3), 441–449 (1993)

    1993

  13. [13]

    Kitada, T.: Multipliers on weighted Hardy spaces over certain totally disconnected groups. Int. J. Math. Math. Sci.11(4), 665–674 (1988)

    1988

  14. [14]

    Kitada, T., Onneweer, C.W.: Weighted Triebel–Lizorkin spaces on locally compact Vilenkin groups. Math. Nachr.168(1), 191–208 (1994)

    1994

  15. [15]

    Acta Math

    Kitada, T., Yang, D.: Potential operators in weighted Herz-type spaces over locally compact Vilenkin groups. Acta Math. Hung.90(1), 29–63 (2001)

    2001

  16. [16]

    Li, W., Xue, Q., Yabuta, K.: Multilinear Calder´ on–Zygmund operators on weighted Hardy spaces. Stud. Math.199, 1–16 (2010)

    2010

  17. [17]

    Lu, G.: Bilinearθ-type Calder´ on–Zygmund operator and its commutator on non-homogeneous weighted Morrey spaces. Rev. Real Acad. Cienc. Exactas Fis. Nat. - A: Mat.115(1), 16 (2021)

    2021

  18. [18]

    Lu, G., Tao, S., Wang, M.: Bilinearθ-type Calder´ on–Zygmund operators and their commuta- tors on product generalized fractional mixed Morrey spaces. Math. Nachr.297(6), 1988–2005 (2024)

    1988

  19. [19]

    Lu, G., Wang, M.: Estimates for bilinearω-type Calder´ on–Zygmund operators and their commutator on the product of grand mixed (generalized) Morrey spaces. Anal. Math.51(3), 941–966 (2025)

    2025

  20. [20]

    Lu, S.Z., Yang, D.C.: The decomposition of Herz spaces on local fields and its applications. J. Math. Anal. Appl.196(1), 296–313 (1995). 23

    1995

  21. [21]

    and Naibo, V.: Weighted norm inequalities for paraproducts and bilinear pseu- dodifferential operators with mild regularity

    Maldonado, D. and Naibo, V.: Weighted norm inequalities for paraproducts and bilinear pseu- dodifferential operators with mild regularity. J. Fourier Anal. Appl.15(2), 218-261 (2009)

    2009

  22. [22]

    Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc.43(1), 126-166 (1938)

    1938

  23. [23]

    Mo, H.X., Han, Z., Yang, L., Wang, X.J.: p-adic singular integrals and their commutators in generalized Morrey spaces. J. Inequal. Appl.2019(1), 65 (2019)

    2019

  24. [24]

    Mo, H., Wang, X., Ma, R.: Commutator of Riesz potential in p-adic generalized Morrey spaces. Front. Math. China13(3), 633–645 (2018)

    2018

  25. [25]

    Nakai, E.: Hardy–Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr.166(1), 95–103 (1994)

    1994

  26. [26]

    Onneweer, C.W., Quek, T.S.: Multipliers on weightedL p-spaces over locally compact Vilenkin groups. Proc. Amer. Math. Soc.105(3), 622–631 (1989)

    1989

  27. [27]

    P´ erez, C., Torres, R.H.: Sharp maximal function estimates for multilinear singular integrals. Contemp. Math.320, 323–332 (2003)

    2003

  28. [28]

    Quek, T.S., Yang, D.: Calder´ on–Zygmund operators on weighted weak Hardy spaces in locally compact Vilenkin groups. Math. Nachr.225(1), 123–143 (2001)

    2001

  29. [29]

    Princeton University Press, Princeton (1975)

    Taibleson, M.H.: Fourier analysis on local fields. Princeton University Press, Princeton (1975)

    1975

  30. [30]

    Wang, C.L.: Variants of the H¨ older inequality and its inverses. Canad. Math. Bull.,20(3), 377-384 (1977)

    1977

  31. [31]

    Wang, L., Shu, L.: Multilinear commutators of singular integral operators in variable exponent Herz-type spaces. Bull. Malays. Math. Sci. Soc.42(4), 1413–1432 (2019)

    2019

  32. [32]

    Wang, W., Xu, J.: Multilinear Calder´ on–Zygmund operators and their commutators with BMO functions in variable exponent Morrey spaces. Front. Math. China12(5), 1235–1246 (2017)

    2017

  33. [33]

    Wang, S., Xu, J.: Weighted norm inequality for bilinear Calder´ on–Zygmund operators on Herz–Morrey spaces with variable exponents. J. Inequal. Appl.2019(1), 251 (2019)

    2019

  34. [34]

    Acta Math

    Yang, D.: Some remarks on singular integrals and power weights on Vilenkin groups. Acta Math. Hung.81(1), 69–88 (1998). 24

    1998